Approaches to Quantum Gravity

422 Questions and answers
– A - R. Gambini and J. Pullin:
Indeed, our method of consistent discretizations using as starting point the
Regge action yields a well defined canonical theory for Regge calculus. The
formulation is equivalent to the original one classically (apart from some
restrictions on the triangulations that are required to have a canonical formulation
that has the same appearance at each point of the manifold). It should
be pointed out that the formulation is canonical but not Hamiltonian, the
evolution is given by a discrete canonical transformation instead of a continuous
time evolution generated by a Hamiltonian. This is reasonable since
Regge calculus discretizes both space and time. An interesting advantage of
the Hamiltonian formulation is that since one naturally restricts the type of
discretizations considered one eliminates the problem of “spikes" and other
pathological structures that may develop in Regge calculus. The disadvantages
include the fact that some of the edges that play the role of Lagrange
multipliers get determined by the evolution equations through complicated
equations that may yield undesired behaviors (like having complex solutions).
In this context the only way of controlling the behavior of these variables is
to choose judiciously the initial data. This type of difficulty has led to the
construction of a special version of consistent discretizations called “uniform
discretizations” where these problems are eliminated. It might be attractive to
pursue Regge calculus with this new approach.
• Q - L. Crane - to J. Henson:
1. It seems one could equally well use a poset to approximate a Lorentzian
manifold in any other dimension than 4. Is there an easy way to put conditions
on a causet so that its dimension doesn’t vary from region to region?
2. In mathematics there are two very different notions of dimension, one topological
and the other measure theoretic. The best known measure theoretical
definition is Hausdorff dimension, which applies to fractals. Do you know of
any approach to differentiating these for causets?
– A - J. Henson:
1. Yes, it is true that causets exist which correspond to manifolds of other
dimensions. It is possible is estimate the dimension of the approximating
manifold, given the causal set alone, and do as the question suggests. By
different regions one might mean different intervals in the causal set, and the
condition that these dimension estimators approximately match, and give the
same value (most interestingly 4) in all regions, is a necessary condition for a
causal set to be “manifoldlike”.
2. At the discrete level, the causal set does not retain the topological or metric
structures of the continuum, which arise at an effective level. So, the question
of comparison only makes sense for causal sets where a continuum